Asymptotic Stability Of Stationary Solution Research Articles

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

Asymptotic stability of stationary solutions for the Kirchhoff equation

This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.

Asymptotic stability of stationary Navier–Stokes flow in Besov spaces

We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.

Pattern formation in a reaction-diffusion-ODE model with hysteresis in spatially heterogeneous environments

This paper is concerned with a reaction-diffusion-ODE system in non-uniform media. The system consists of a diffusive species and a non-diffusive species and models biological pattern formation. We prove the existence of (i) stationary solutions for all values of the diffusion coefficient D by applying a generalized mountain pass lemma, (ii) stationary solutions with transition layer for D sufficiently small by applying the singular perturbation method, and (iii) monotone stationary solutions for D sufficiently large by the upper and lower solution method. Also, the asymptotic stability of stationary solutions is studied.

Asymptotic stability of stationary solutions to the drift-diffusion model with the fractional dissipation

We study the drift-diffusion equation with fractional dissipation $$(-\varDelta )^{\theta /2}$$ arising from a model of semiconductors. First, we prove the existence and uniqueness of the small solution to the corresponding stationary problem in the whole space. Moreover, it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data are suitably close to the stationary solution and the stationary solution is sufficiently small.

Stability and Existence of Stationary Solutions to the Euler–Poisson Equations in a Domain with a Curved Boundary

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of walls immersed in a plasma, and to analyze qualitative information of such a sheath layer. In the case of planar wall, Bohm proposed a criterion on the velocity of the positive ion for the formation of sheath, and several works gave its mathematical validation. It is of more interest to analyze the criterion for the nonplanar wall. In this paper, we study the existence and asymptotic stability of stationary solutions for the Euler-Poisson equations in a domain of which boundary is drawn by a graph. The existence and stability theorems are shown by assuming that the velocity of the positive ion satisfies the Bohm criterion at infinite distance. What most interests us in these theorems is that the criterion together with a suitable necessary condition guarantees the formation of sheaths as long as the shape of walls is drawn by a graph.

Stability of the radially symmetric stationary wave of the Burgers equation with multi‐dimensional initial perturbations in exterior domain

AbstractThe present paper is concerned with stability of the stationary solution of the Burgers equation in exterior domains in . In the previous papers [5, 6, 7] the asymptotic behavior of radially symmetric solutions for the multi‐dimensional Burgers equation in exterior domains in , has been considered. The results [5, 6, 7] are restricted to stability of radially solutions within the class of spherically one dimensional flow. However, from a viewpoint of fluid dynamics, it is the rare case that such a radially symmetric stationary wave remains to be a radial flow under the initial disturbance. Hence it seems to be natural to handle the non‐radially symmetric perturbed fluid motion even from the radially symmetric one. On the other hand, Kozono and Ogawa [8] showed the asymptotic stability of stationary solutions for the incompressible Navier–Stokes equation on multi‐dimensional spaces. In this paper we apply their method [8] to the multidimensional Burgers equation, and show the asymptotic stability for stationary wave on .

Asymptotic stability of stationary waves to the Navier–Stokes–Poisson equations in half line

ABSTRACT The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier–Stokes–Poisson equations with the classical Boltzmann relation in a half line. We first show the unique existence of stationary solution with the aid of the stable manifold theory, and then prove that the stationary solution is time asymptotically stable under the small initial perturbation by the elementary energy method. Finally, we discuss the convergence rate of the time-dependent solution towards the stationary solution, and give a new condition to ensure an algebraic decay or an exponential decay. The proof is based on a time and space weighted energy method by fully utilizing the self-consistent Poisson equation.

The mathematics of asymptotic stability in the Kuramoto model.

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.

Stability preservation in Galerkin-type projection-based model order reduction

We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalize this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.

On artifact solutions of semi-analytic methods in nonlinear dynamics

Nonlinear dynamics is a topic of permanent interest in mechanics since decades. The authors have recently published some results on a very classical topic, the dynamics of a softening Duffing oscillator under harmonic excitation focusing especially on low-frequency excitation (von Wagner in Arch Appl Mech 86(8):1383–1390, 2016). In this paper, it was shown that classical tools like harmonic balance and perturbation analysis may produce artificial solutions when applied without extra carefulness with respect to parameter ranges in the case of perturbation analysis or prior knowledge about the type of solution in case of harmonic balance. In the present paper these results are shortly summarized as they give the starting point for the additional investigations described herein. First, the method of slowly changing phase and amplitude is reviewed with respect to its capability of determining asymptotic stability of stationary solutions. It is shown that this method can also produce artifact results when applied without extra carefulness. As next example an extended Duffing oscillator is investigated, which shows, if harmonic balance is applied, “islands” of solutions. Using the error criterion in harmonic balance as described in von Wagner (2016) again artifact solutions can be identified.

Asymptotic stability of stationary solutions for Hall magnetohydrodynamic equations

In this paper, we consider the large time behavior of the compressible Hall magnetohydrodynamic equations with Coulomb force in $$\mathbb {R}^3$$ near the non-constant equilibrium state. We derive the global existence provided that the initial perturbation is sufficiently small. Moreover, under the further assumption that the doping profile is of small variation, we obtain the convergence rates by combining the linear $$L^p$$ – $$L^q$$ decay estimates.

Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions

This paper is concerned with spherically symmetric motions of non-isentropic viscous gaseous stars with self-gravitation. When the stationary entropy S‾(x) is spherically symmetric and satisfies a suitable smallness condition, the existence and properties of the stationary solutions are obtained for 65<γ<2 with weaker constraints upon S‾(x) compared with the one in [26], where γ is the adiabatic exponent. The global existence of strong solutions capturing the physical vacuum singularity that the sound speed is C12-Hölder continuous across the vacuum boundary to a simplified system for non-isentropic viscous flow with self-gravitation and the nonlinear asymptotic stability of the stationary solution are proved when 43<γ<2 with the detailed convergence rates, motivated by the results and analysis of the nonlinear asymptotic stability of Lane–Emden solutions for isentropic flows in [29,30].

On the Asymptotic Stability of Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

We study the stability of stationary solutions of the two dimensional inviscid incompressible porous medium equation (IPM). We show that solutions which are near certain stable stationary solutions must converge as t → ∞ to a stationary solution of the IPM equation. It turns out that linearizing the IPM equation about certain stable stationary solutions gives a non-local partial damping mechanism. On the torus, the linearized problem has a very large set of stationary (undamped) modes. This makes the problem of long-time behavior more difficult since there is the possibility of a cascading non-linear growth along the stationary modes of the linearized problem. We solve this by, more or less, doing a second linearization around the undamped modes, exploiting a special non-linear structure, and showing that the stationary modes can be controlled.

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L2‐decay estimates for the linearized equations. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Asymptotic stability of stationary solutions for magnetohydrodynamic equations

In this paper, we are concerned with the compressible magnetohydrodynamic equations with Coulomb force in three-dimensional space. We show the asymptotic stability of solutions to the Cauchy problem near the non-constant equilibrium state provided that the initial perturbation is sufficiently small. Moreover, the convergence rates are obtained by combining the linear Lp-Lq decay estimates and the higher-order energy estimates.

On the Asymptotic Stability of Steady Flows with Nonzero Flux in Two-Dimensional Exterior Domains

The Navier-Stokes equations in a two-dimensional exterior domain are considered. The asymptotic stability of stationary solutions satisfying a general hypothesis is proven under any $L^2$-perturbation. In particular the general hypothesis is valid if the steady solution is the sum of the critically decaying flux carrier with flux $|\Phi| < 2\pi$ and a small subcritically decaying term. Under the central symmetry assumption, the general hypothesis is also proven for any critically decaying steady solutions under a suitable smallness condition.

The stability of stationary solution for outflow problem on the navier-stokes-poisson system

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.

Stability of stationary solutions for the non-isentropic Euler-Maxwell system in the whole space

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R3. It is known in the authors’ previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.

Asymptotic stability of stationary solutions to the compressible Euler-Maxwell equations

In this paper, we are concerned with the compressible Euler-Maxwell equations with a nonconstant background density (e.g. of ions) in three dimensional space. There exist stationary solutions when the background density is a small perturbation of a positive constant state. We first show the asymptotic stability of solutions to the Cauchy problem near the stationary state provided that the initial perturbation is sufficiently small. Moreover the convergence rates are obtained by combining the $L^p$-$L^q$ estimates for the linearized equations with time-weighted estimate.